Hamburger Synchrotronstrahlungslabor HASYLAB at Deutsches Elektronen Synchrotron DESY. Notkestr. 85, Hamburg, Germany

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1 Magnet Sorting for the TTF-FEL Undulator using Simulated Annealing B. Faat and J. Puger Hamburger Synchrotronstrahlungslabor HASYLAB at Deutsches Elektronen Synchrotron DESY Notkestr. 85, 223 Hamburg, Germany Abstract In order to get the eld quality required for the undulator of the TESLA Test Facility Free Electron Laser (TTF-FEL), two steps have been foreseen. The rst step is to sort the magnets in such away that the resulting eld quality is as good as possible with the magnets at hand. The second step is to use techniques such as pole height adjustment and/or shimming to tune both x and y eld components to the required specications. The main purpose of the magnet sorting is to minimie corrections in the second step. Therefore, two transverse components of the eld, ideally ero, are minimied by this procedure. In this paper we outline the magnet sorting procedure using simulated annealing. For the three undulator segments for phase I of the TTF-FEL geometrical constrains were properly taken into account. The results obtained for the magnets being used are presented. 1 Introduction The goal of the TTF-FEL [1] is to supply future users with VUV radiation, down to a wavelength of 6 nm. It will operate as a single pass device starting from noise, the so-called Self Amplied Spontaneous Emission (SASE) regime [2,3]. The electron beam will be supplied by a low emittance photo cathode gun and accelerated by a high-gradient superconducting accelerator. In order to get the desired peak current of 2.5 ka, the electron bunches will be compressed in three stages. The radiation will be produced in an undulator with a superimposed alternating gradient (FODO) focusing structure. In order to test these components at an early stage and prove the SASE principle in this wavelength range, rst tests will be performed with only three accelerating modules in place (E b < 39 MeV), two bunch compressors, resulting in a peak current of 5 A, and half of the undulator consisting of three segments. Parameters are given in Table 1. In order for the radiation to reach saturation within the given undulator length, we have to guarantee a close overlap between electron beam and radiation eld. Earlier simulations have shown that this can be achieved if the electron beam does not deviate from its ideal trajectory by more than % of its rms beam sie, i.e. 1 to 12 m in case of the TTF-FEL [4]. Although corrector stations are foreseen along the entire undulator to correct possible errors, one should still make the undulator quality as good as possible in order to keep corrections small.

2 Table 1 Undulator and electron beam parameters for the TTF FEL (Phase I). Electron beam Peak current Normalied rms emittance rms energy spread average beam sie undulator number of segments 3 length of segment period length undulator peak eld length of quadrupoles number of quads./segment 1 distance between quads. quad. gradient 5 A 2 mm mrad 5 kev 7 m m 27.3 mm.497 T mm 341 mm 12.5 T/m Undulators are manufactured from individual parts like permanent magnets, soft iron poles, holders etc. which cannot be produced to be absolutely perfect. Mechanical parts contain dimensional errors typically in the order of to 5 m. By way the largest error source is the magnetic material itself. Its magnetiation typically diers by 1% and the angle of magnetiation may vary by as much as 2 ; 2:5.Furthermore the material can be inhomogeneous. In order to obtain a perfect undulator eld two steps seem reasonable. In the rst step, the inuence due to imperfect magnets is minimied by sorting magnets in such a way that errors from dierent magnets mutually cancel. To do so each magnet has to be measured and characteried. The magnetiation, angular deviation from the nominal direction and its homogeneity is measured and characteried by a set of numbers. These numbers are input in a sorting algorithm using the method of simulated annealing [5{7]. The algorithm exchanges and ips magnets to optimie a goal function which has previously been dened and will be described in detail below. This paper deals with the sorting of the magnets for the undulator for the phase I at the TESLA Test Facility consisting of three undulator segments. Each of them is m long and consists of 652 magnets that have to be sorted. For the second step of optimiation, pole height adjustment and shimming will be used to compensate remaining errors. These issues are not subject to this paper and have been presented elsewhere [8]. 2 Procedure for magnet sorting The main problem in sorting the magnets is their large number. For the TTF-FEL undulator structure, each undulator consists of 652 magnets. Checking all possible congurations would clearly be impossible. Therefore, we use a method referred to as simulated annealing. This method uses random changes in the order in which magnets are placed. Every change made is checked against a predened function which is to be minimied. The whole TTF-FEL phase I undulator consists of 3 segments with 652 identical magnets each that have to be sorted. Consequently, all magnets can be placed at an arbitrary position in any of the three undulator segments. For reasons of logistics (keeping track of all magnets while actually building the undulator) it has been decided to divide the total number of magnets available (22 in this case) in three equal parts, optimiing each undulator using only one third of the magnets. Although this limits the possible con- gurations, the desired accuracy can still be achieved, as will be shown in this report. At a specic position in the undulator, the eld direction of the magnet is given (see Fig. 1). At this position, the magnet can only be rotated by 18 degrees around the axis 2

3 Permanent- Magnet Pol in the direction of the main eld. This rotation will be referred to as a "FLIP" of the magnet. Flipping a magnet is the only operation that can be performed on a single magnet without changing its location. One can also interchange the position of two magnets. Since at each position the main eld direction is given, this operation is unique except for a possible ip of one or both magnets. However, since it is always possible to ip afterwards, this has no consequence. For convenience, it is assumed that a ip is connected to a position rather than to a magnet. Note, that since in general there are more magnets available than needed, one can also change a magnet inside the undulator with one that was not used so far. Consequently, the nal magnet distribution will not include magnets which have the worst quality. To summarie, there are two possible (fundamental) operations possible, namely ipping one magnet or changing the position of twomagnets.from a given distribution of magnets, the undulator quality can sometimes only improve after two or three of the above mentioned operations. The advantage of simulated anneal is that it allows the quality to get worse within given boundaries. A number of subsequent changes is abandoned only if the quality does not improve after several operations. The procedure starts again from the best undulator quality thus far. Although the nal solution does not always give the absolute best undulator quality, results will show that the improvement is adequate for the TTF-FEL undulator. y x Gap e Fig. 1. Layout of the TTF-FEL undulator. Fig. 2. Labeling and block coordinate system for the magnets. Indicated are the three magnetiation directions and the two magnetic eld measurements. The magnet block sie is 7 by 5by 3.65 mm. Each magnet is speci- ed by a magnet label, a unique number for each magnet, and an orientation to specify between the normal and ipped orientation (see text for details). Each block magnet has three magnetiation components, M x, M y and M (see Fig. 2 for details). These components are measured for each magnet individually using a Helmholt coil setup. For the -direction, two additional measurements are made, namely B (n) and B (s), attwo geometrically equivalent positions near the north and south pole using a Hall probe mounted in a suitable xture, in order to take into account an inhomogeneous part of the magnetiation. For an ideal magnet, M x and M y are equal to ero and B (n) = B (s). The average value of the Hall probe measurements is assumed to correspond to the measured magnetiation of the magnet, i.e. B (n) B (s) / M (n) = / M (s) = M (1 + B (s) =B (n) ) M and (1a) (1 + B (n) =B (s) ) : (1b) It is easily seen that B (n) + B (s) / M. Similarly, it is assumed that B (x y) / M (x y). 3

4 From these four values, elds can now be calculated per half period. Each magnet pair at position i, one of them in the lower (`) structure and in the upper (u), results in the following elds B x i / (F` i M x ` i + F u i M x u i )(;1) i : B y i / F` i M y ` i ; F u i M x u i : B (1) und i / (M (n) ` i + M (s) u i )(;1)i;1 : B (2) und i / (M (s) ` i + M (n) u i )(;1)i : The factor F stands for a possible ip of the magnet at this position. The uxlines of the main magnetiation M are going through the poles between the magnets, resulting in the undulator eld in the y-direction. The superscripts (1) or (2) stand for the two halfs of the magnet pair. The pairs (1) (or (2)) at position i have to be added to (2) (or (1)) at position i + 1 to get the eld at the pole position between the two magnets. What combination has to be added depends on the position within the undulator. Now that all elds are given at the magnet positions (B x and B y ) and at the poles (B und ), one is able to calculate the function that has to be optimied. One of the problems is that the proportionality constant between magnetiation M and eld B is unknown and not neccessarily the same for the dierent eld components. The magnetic eld consists of an x-component and a y-component, of which the latter is an addition of the direct eld (ycomponent of the magnetiation of two magnets) and the undulator eld (either B (1) B (2) und i+1 und i + or B(2) B(1) und i und i+1 ). How to add the two sources to the y-component is unknown. Therefore, instead of minimiing the x and y- component of a function, three components are minimied independently, e.g. two components in x and y-direction, and the third component which is related to the desired undulator eld (denoted as w for `wiggler' in the remainder of this report). The question to be answered is what function has to be minimied, i.e. how to specify the undulator quality. The most common way to determine the quality is to specify the rms value of the eld errors, i.e. add the squares of the deviation of the peak eld every half period compared to the average eld. Simulations have shown, however, that this is a very poor parameter with which to determine the quality [1,11]. The main parameters that determine the performance of an undulator are the phase shake and the second eld integral [4,11]. The function that has been optimied in the report is therefore the second eld integral. The phase shake is expected not to be a signicant problem for these small magnetic eld errors. 3 General layout of the program The program used to determine the optimum conguration consists of several parts. Obviously it has to determine the input and output le names as well as some parameters to determine what undulator segment has to be sorted etc. Both input and output le have the same format, consistent with the information given by J. Pueger [9]. The reason for this is that one possibly needs an additional optimiation, either because the undulator quality is still not adequate, but also because some magnets can be damaged during the implementation into the structure, in which case some magnets have to be re-ordered. For this reason, one can specify if certain magnets have to be excluded from the sorting procedure. Although one could in this case in principle completely sort the magnets again according to the same procedure, it has been found that usually it is sucient to change the magnet which is no longer available by another magnet that has not yet been used. Checking only the remaining magnets (of the order of ) in two possible orientation is very fast in the tests performed and so far did not increase 4

5 the test function (e.g. rms eld error or second eld integral) signicantly. The les contain a magnet number with its three magnetiation components and twovalues of a magnetic induction, one value at the south pole and one at the north pole. In addition the position of the magnets within the total structure is given. (1) Undulator segment number (1-3) (2) Upper/Lower (U/L)part (3) Module number within a segment (1-5) (4) Magnet position within each module (1-65) 1 (5) Magnet identication number (6) Flipped/Normal (F/N) magnet The rst four numbers uniquely determine the magnet position within the structure, the fourth number determines which magnet is at the specied position and the last number determines the orientation of the magnet. The core of the program consists of a few parts. One procedure determines by invoking a random number generator which of the two operations, ipping a magnet or interchanging two magnets, has to be performed. It also determines, again using random numbers, at which position a magnet has to be ipped or at what positions magnets have to be interchanged. A second procedure performes the actual operations, i.e., ipping the magnet at the position or changing magnets at two dierent positions. In addition, the magnetic elds are changed to take into account the changed magnets. The next procedure checks the second eld integral in three directions, x, y and w (where w is the main magnetic eld, see the previous section). The main loop of the program now determines if the quality has been improved and if not, if the new con- guration has to be excepted anyway. This procedure continues until no signicant im- 1 The last magnet pair (half an undulator period) in each undulator is given separately provement is expected, e.g. no improvement has occurred after many changes of magnet position and orientation. 4 Results of simulations for the TTF-FEL undulator For undulator I, a total of 674 magnets are available. The distribution of x y and - components of the measured ux for these magnets is given in Fig. 3. One can also see Number of magnets Mean = sd = 9.61 Mean = 15.9 sd = Mean = sd = corresponds to ero (a) Magnetic Flux (1-5 ) Fig. 3. Distribution of the magnetic ux for the 7 magnets available for the rst undulator. (a) gives the ux for the main eld component, (b) for the magnetiation in the x-direction, (c) for the y-direction. Note that these distributions are not independent, but coupled to each other. that the average of the two transverse components, x and y, is not equal to ero. This has been checked carefully with dierent Helmhol coils to exclude systematic errors. (b) (c) 5

6 The asymmetric distribution of the elds becomes more apparent in Fig. 4. The larger Number of magnets Mean = E-5 sd = Mean = E-5 sd =.5458 Mean = -5.57E-5 sd =.999 (a) Magnetic field error Fig. 4. Distribution of the magnetic eld for the rst 326 half-periods of the initial conguration before sorting, normalied to the average undulator eld. The top gure (a) gives the eld for the main component, the middle gure (b) for the eld in the x-direction, the bottom gure (c) for the y-direction. After redirection of the main eld ux through the poles, this eld will be directed in the y-direction. width in the x-direction is due to the fact that the distribution consists of two Gaussians. The electron trajectories due to these elds are shown in Fig. 5. The rms-value of the second eld integral is given in the gure (where w rms stands for the main undulator eld). If one includes the rst magnet pair to compensate the second eld integral, hence having the electron beam on axis at the end of the undulator, all rms values become slightly less than 5 m. From the beam trajectories obtained with the unsorted magnets shown in Fig. 5 it is obvi- (b) (c) Beam position (µm) uncorrected beam w rms = 252 µm x rms = 111 µm y rms = 25 µm (m) Fig. 5. Electron beam trajectories due to the eld components in Fig. 4. Calculations have been performed at an energy of 225 MeV, the minimum energy foreseen for phase I of the project. w rms indicates the trajectory due to the deviation of the measured eld compared to the ideal undulator eld, the other two components due to the remaining error elds. ous that some sorting is neccessary in order to keep the beam on axis. Results of sorting after optimiing the second eld integral are shown in Fig. 6. Note the angle of the undulator trajectory. This is due to the fact that the rst magnet was not included in the optimiation procedure of the main undulator eld. Since adjustable magnets are foreseen to correct this overall initial kick the rms value is comparable to the other two. If instead of the second eld integral the rms-eld error would have been used as optimiation criterium, the resulting second eld integral would be approximately an order of magnitude larger, even if the rst magnet pair is used for compensation. For the second undulator, also the rst magnet has been included in the minimiation procedure. Results are shown in Fig. 7. One should note the following. Because of the distribution of the eld in the x-direction (see Fig. 3 for the corresponding ux), one basically looses a degree of freedom. Normally, in order to get a total x-eld equal to ero, one can choose two magnets with the same x-ux or opposite x-ux. Since the distribution is not centered around ero, the second possibil- 6

7 Beam position (µm) second field integral corrected w rms = 6.82 µm x rms =.56 µm y rms =.29 µm (m) Fig. 6. Electron beam trajectories after minimiing the second eld integral. Calculations have been performed at an energy of 225 MeV, the minimum energy foreseen for phase I of the project. w rms indicates the trajectory due to the deviation of the measured eld compared to the ideal undulator eld, the other two components due to the remaining error elds. ity does no longer exist. Therefore, in order to minimie the x-component of the magnetic eld, one has to choose two magnets with the same ux. In this case, a ip of one magnet is no longer possible. Therefore, both magnets also need to have opposite y-components, if that is to be minimied simultaneously. 5 Conclusions & Limitations As has been shown, magnet sorting greatly improves the quality of the undulator. The simulated beam trajectories show the deviation from the ideal of only a few micrometers. This value is more than enough for the TTF- FEL, and would even be enough for the future TESLA-FEL, where the acceptable trajectory deviation goes down from 12 m rms to typically 3 m. Beam position (µm) second field integral corrected (second undulator) w rms =.26 µm x rms =.75 µm y rms =.51 µm (m) Fig. 7. Electron beam trajectories of the second undulator after minimiing the second eld integral. Calculations have been performed at an energy of 225 MeV, the minimum energy foreseen for phase I of the project. w rms indicates the trajectory due to the deviation of the measured eld compared to the ideal undulator eld, the other two components due to the remaining error elds. A few remarks should be made however. In calculating magnetic elds, it has been assumed that they are proportional to the magnetiation of the individual magnet blocks. This assumption is no more than a rst estimated guess of the actual eld produced by the magnets. For example, the poles between the magnets have been assumed to be ideally placed and 1 % conducting. Hence, errors that might be caused by the poles have not been taken into account. Furthermore, the beam trajectory calculated assumes halfcosine elds per half period. The actual eld is more complicated. In addition, coupling of the B y eld with the undulator eld has not been taken into account. The elds have been treated completely independently. However, if we assume that the poles can be adjusted accurately enough for the (undulator) eld errors to become small, it is reasonable to assume that the direct elds will give a negligibly small eect. Thus the main contribution to the trajectory deviation will most certainly be the quadrupole misalignment, which has 7

8 not been taken into account in any of the sorting procedures so far, even when they can be aligned with an accuracy of 1 m. As stated in the introduction, the eld errors dominate the geometrical errors. This is probably no longer true after the magnets have been sorted. Therefore, the actual electron beam trajectories might be signicantly dierent from those calculated assuming only the magnetic errors. After the elds have actually been measured, one should check if there is a strict relation between simulated and measured second eld integrals. With this new knowledge, perhaps a renement of the method can be obtained. References [1] \A VUV Free Electron Laser at the TESLA Test Facility: Conceptual Design Report", DESY Print TESLA-FEL 95-3, Hamburg, DESY, 1995 [2] A.M. Kondratenko and E.L. Saldin, Generation of Coherent Radiation by a Relativistic Electron Beam in an Ondulator, Particle Accelerators, 1 (198) 7. [3] R. Bonifacio, C. Pellegrini and L.M. Narducci Optics Commun. 53 (1985) 197 [6] S. Kirkpatric, C.D. Gelatt, Jr. M.P. Vecchi, Optimiation by Simulated Annealing, Science 2 (1983) 671. [7] Numerical Recipes, William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery, Cambridge University Press, 436. [8] J. Puger, H. Lu and T. Teichmann, Field ne tuning by pole height adjustment for the undulator of the TTF-FEL, Presented at the th International FEL conference, August 17-21, 1998, Williamsburg, USA [9] J. Puger, Protokoll der 1. Projektbesprechung VUV-FEL Undulator I, Internal note. [1] B.L. Bobbs, G. Rakowsky, P. Kennedy, R.A. Cover and D.Slater In Search of a Meaningful Field-Error Specication for Wigglers Nucl. Instr. Meth. 296 (199) [11] Yu.M. Nikitina and J. Puger Inuence of Magnetic eld errors on the spectrum of the Petra Undulator Nucl. Instr. Meth. A359 (1995) [12] J. Puger, H. Lu, D. Koster and T. Teichmann, Magnetic Measurements on the Undulator prototype for the VUV- FEL at the TESLA Test Facility Nucl. Instrum. Meth. A7 (1998) 386. [4] B. Faat, J. Puger and Yu.M. Nikitina, Study of the undulator specication for the VUV-FEL at the TESLA Test Facility, Nucl. Instrum. Meth. A393 (1997) 38. [5] Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, Edward Teller, Equation of State Calculations by Fast Computing Machines, The Journal of Chemical Physics 21 (1953)